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Chapter 4: Problem 27

Differentiate. $$ f(x)=\frac{e^{x}}{x^{4}} $$

Short Answer

Expert verified
\( \left( \frac{e^x}{x^4} \right)' = \frac{e^x (x - 4)}{x^5} \)

Step by step solution

01

Identify the Rule

Recognize that the function can be differentiated using the quotient rule, which is stated as: If \( u(x) \) and \( v(x) \) are differentiable functions, then the derivative of their quotient \( \frac{u(x)}{v(x)} \) is given by: \[ \left(\frac{u(x)}{v(x)} \right)' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \]
02

Identify u(x) and v(x)

Identify the numerator and the denominator of the function. Here, \( u(x) = e^x \) and \( v(x) = x^4 \).
03

Differentiate u(x)

Differentiate \( u(x) = e^x \). The derivative of \( e^x \) is \( u'(x) = e^x \).
04

Differentiate v(x)

Differentiate \( v(x) = x^4 \). Using the power rule, the derivative of \( x^4 \) is \( v'(x) = 4x^3 \).
05

Apply the Quotient Rule

Substitute all the values into the quotient rule formula: \[ \left( \frac{e^x}{x^4} \right)' = \frac{e^x \( x^4 \) - e^x \( 4x^3 \)}{(x^4)^2} \]
06

Simplify the Expression

Simplify the numerator and the denominator: \[ \left( \frac{e^x}{x^4} \right)' = \frac{e^x (x^4 - 4x^3)}{x^8} \] Factor out \( x^3 \) in the numerator: \[ \left( \frac{e^x}{x^4} \right)' = \frac{e^x x^3 (x - 4)}{x^8} \] Further simplify the fraction: \[ \left( \frac{e^x}{x^4} \right)' = \frac{e^x (x - 4)}{x^5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a central branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It examines how things change and enables us to describe motion and changing systems. Calculus is divided into two main branches:
  • Differential Calculus, which focuses on the concept of the derivative and how functions change.
  • Integral Calculus, which deals with the accumulation of quantities and the areas under and between curves.
Such tools are essential for solving complex problems in physics, engineering, economics, and more.
Derivative
A derivative represents how a function changes as its input changes. In mathematical terms, it gives the slope of the tangent line to the function's graph at a particular point. For a function f(x), the derivative is denoted by f'(x) or \( \frac{d}{dx}f(x) \). There are several notations for derivatives, but they essentially mean the same thing: the rate of change of the function with respect to its variable:
\[ f'(x) = \frac{d}{dx} f(x) \]
We use derivatives to solve problems involving rates of change and to find maximum and minimum values of functions.
Differentiation Techniques
Differentiation techniques are methods for finding derivatives. Key techniques include:
  • Power Rule: Useful for functions of the form \( x^n \). If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
  • Product Rule: For the product of two functions \( u(x) \) and \( v(x) \), the derivative is \( u'v + uv' \).
  • Quotient Rule: For the quotient of two functions \( u(x) \) and \( v(x) \), the derivative is: \[ \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \]
  • Chain Rule: For a composite function \( g(f(x)) \), the derivative is \( g'(f(x)) \times f'(x) \).
These rules allow us to tackle more complex functions by breaking them into simpler parts.
Exponential Functions
Exponential functions are mathematical functions of the form \( f(x) = a \times e^{bx} \) where \( e \) is the base of the natural logarithm. A common exponential function is \( e^x \), where the derivative is the same as the original function: \( \frac{d}{dx}e^x = e^x \). This unique property makes exponential functions particularly important in calculus. They model many natural phenomena, such as population growth, radioactive decay, and continuous compound interest.
Power Rule
The Power Rule is one of the simplest and most widely used differentiation techniques. It applies to any function of the form \( x^n \), where \( n \) is any real number. The rule states:
\[ \frac{d}{dx}x^n = nx^{n-1} \]
For example, if \( f(x) = x^4 \), then \( f'(x) = 4x^3 \). This rule simplifies differentiation for polynomial functions and is foundational for more advanced differentiation techniques. By mastering the Power Rule, students can easily handle more complex algebraic expressions.

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Most popular questions from this chapter

Suppose that \(P_{0}\) is invested in a savings account in which interest is compounded continuously at \(8 \%\) per year. That is, the balance \(P\) grows at the rate given by $$ \frac{d P}{d t}=0.08 P $$ a) Find the function that satisfies the equation. List it in terms of \(P_{0}\) and \(0.08\). b) Suppose that \(\$ 20,000\) is invested. What is the balance after \(1 \mathrm{yr} ?\) after \(2 \mathrm{yr} ?\) c) When will an investment of \(\$ 20,000\) double itself?

In a chemical reaction, substance \(A\) decomposes at a rate proportional to the amount of \(A\) present. a) Write an equation relating \(A\) to the amount left of an initial amount \(A_{0}\) after time \(t\). b) It is found that \(8 \mathrm{~g}\) of \(A\) will reduce to \(4 \mathrm{~g}\) in \(3 \mathrm{hr}\). After how long will there be only \(1 \mathrm{~g}\) left?

Acidity. Pure water is neutral with a pH of \(7 .\) a) What is the concentration of the hydronium ions in pure water? b) Suppose that during an experiment the concentration of the hydronium ions is given by \(x=0.001 t+10^{-7}\), where \(0 \leq t \leq 100\) is the time measured in seconds. Find a [ormula relating the \(\mathrm{pH}\) of the solution to the time \(t\). c) Use your answer to part (b) to compute the rate of change of the \(\mathrm{pH}\) of the solution. d) What is the \(\mathrm{pH}\) at time \(0 ?\) e) At what time does the \(\mathrm{pH}\) of the solution change most rapidly? What is the \(\mathrm{pH}\) of the solution at that time?

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